Which of the following is an irrational number?

Study for the 8th Grade Mathematics Test. Prepare with multiple choice questions, each with hints and explanations. Get ready for your exam!

Multiple Choice

Which of the following is an irrational number?

Explanation:
Rational versus irrational numbers: an irrational number cannot be written as a ratio of integers, and its decimal expansion goes on forever without repeating. Among the options, the first three can be written as fractions of integers or are integers, so they’re rational, and their decimals terminate: 0 is 0.0, 1/2 is 0.5, and 3/4 is 0.75. The square root of 2 cannot be expressed as a fraction of integers; its decimal goes on without repeating. A classic proof shows that if √2 were a fraction a/b in lowest terms, then a^2 would equal 2b^2, forcing both a and b to be even, which contradicts the assumption that the fraction was in lowest terms. So √2 is irrational, which is why it’s the correct choice.

Rational versus irrational numbers: an irrational number cannot be written as a ratio of integers, and its decimal expansion goes on forever without repeating. Among the options, the first three can be written as fractions of integers or are integers, so they’re rational, and their decimals terminate: 0 is 0.0, 1/2 is 0.5, and 3/4 is 0.75. The square root of 2 cannot be expressed as a fraction of integers; its decimal goes on without repeating. A classic proof shows that if √2 were a fraction a/b in lowest terms, then a^2 would equal 2b^2, forcing both a and b to be even, which contradicts the assumption that the fraction was in lowest terms. So √2 is irrational, which is why it’s the correct choice.

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